9 MEASURABLE FUNCTIONS 135 The sets obtained from the open (or closed) sets of X by iterated application of the operations (i) and (ii) above are therefore also universally measurable. The definition of a ^-measurable set A also shows that there exists a uni- versally measurable set B contained in A such that A — B is /^-negligible. By applying this result to X — A, we see that there also exists a universally measurable set C containing A such that C — A is /(-negligible. A mapping u of X into a topological space Y is said to be measurable (with respect to ju) or ^-measurable if there exists a partition of X into a sequence of compact sets Kn and a ^-negligible set N such that each of the restrictions u \ Kn is continuous. The mapping u is said to be universally measurable if it is measurable for all measures on X. A continuous function is universally measurable, For a subset A of X to be measurable (resp. universally measurable) it is necessary and sufficient that its characteristic function (pA should be measur- able (resp. universally measurable). For if we have a partition of X consisting of a sequence of compact sets Kn and a negligible set N, such that <pA | Kn is continuous for all n, then Kn is the union of two disjoint compact subsets, namely A n Kn and KM - (A n KJ ((3.11.4) and (3.17.3)), and A is the union of the sets A n Kn and A n N, and is therefore measurable. Conversely, if A is measurable, then so is X — A (13.9.3), hence there exists a partition of X consisting of a sequence of compact sets Kn and a negligible set N, such that either KM c A or Kn c X — A for all n. Consequently <pA | Kn is continuous for all n. (1 3.9.4) Let u be a mapping ofX. into a topological space Y. Then the following three properties are equivalent: (a) u is ^.-measurable. (b) For each compact subset K of X and each s > 0, there exists a com- pact subset K' of K such that ^(K — K') ^ £ and such that the restriction of u to K' is continuous. (c) For each compact subset K of X, there exists a partition ofK. consisting of a sequence (LM) of compact sets and a ^-negligible set M, such that each of the restrictions u \ Ln is continuous. To show that (a) implies (c), we remark that the hypothesis (a) implies the existence of a partition of X consisting of a sequence (KJ of compact sets and a negligible set N, such that u \ Kn is continuous for all n ; hence (c) is satisfied by taking LM = K n Kn and M = K n N. To show that (c) implies (b), we remark that it follows from (c) that (13.8.7), and hence that there is an integer n such that n-l /i(K — K') g e, where K' = (J Lt . Clearly K' satisfies the conditions of (b).serve that there exists a se-